论文标题
具有积分RICCI曲率边界的度量测量空间的稳定性
Stability of metric measure spaces with integral Ricci curvature bounds
论文作者
论文摘要
在本文中，我们研究了稳定性和紧凑性W.R.T.具有整体RICCI曲率边界的平滑度量测量空间的Gromov-Hausdorff收敛。更准确地说，我们证明了一系列$ n $维的riemannian流形的子对度度量空间，该量度满足了lottm-sturm-villani的意义上满足曲率差异条件$ cd（k，n）$，从$ k $收敛到$ 0 $。结果也适用于一般平滑度量度量空间的序列$（m，g_m，e^{ - f} \ mbox {vol} _m）$，其中bakry-emery曲率取代了RICCI曲率。推论是Brunn-Minkowski-Type不平等，帽子的估计值和关于基本群体有限的声明。加上统一的非策略条件，极限甚至满足了Riemannian曲率维度条件$ rcd（k，n）$。这意味着数量和直径几乎刚性定理。
In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of $n$-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition $CD(K,n)$ in the sense of Lott-Sturm-Villani provided the $L^p$-norm for $p>\frac{n}{2}$ of the part of the Ricci curvature that lies below $K$ converges to $0$. The results also hold for sequences of general smooth metric measure spaces $(M,g_M, e^{-f}\mbox{vol}_M)$ where Bakry-Emery curvature replaces Ricci curvature. Corollaries are a Brunn-Minkowski-type inequality, a Bonnet-Myers estimate and a statement on finiteness of the fundamental group. Together with a uniform noncollapsing condition the limit even satisfies the Riemannian curvature-dimension condition $RCD(K,N)$. This implies volume and diameter almost rigidity theorems.
